Dynamics of polarization-tuned mirror symmetry breaking in a rotationally symmetric system

Lateral momentum conservation is typically kept in a non-absorptive rotationally symmetric system through mirror symmetry via Noether’s theorem when illuminated by a homogeneous light wave. Therefore, it is still very challenging to break the mirror symmetry and generate a lateral optical force (LOF) in the rotationally symmetric system. Here, we report a general dynamic action in the SO(2) rotationally symmetric system, originating from the polarization-tuned mirror symmetry breaking (MSB) of the light scattering. We demonstrate theoretically and experimentally that MSB can be generally applied to the SO(2) rotationally symmetric system and tuned sinusoidally by polarization orientation, leading to a highly tunable and highly efficient LOF (9.22 pN/mW/μm−2) perpendicular to the propagation direction. The proposed MSB mechanism and LOF not only complete the sets of MSB of light-matter interaction and non-conservative force only using a plane wave but also provide extra polarization manipulation freedom.


Supplementary Note 1: The LOF for an SO(2) rotationally symmetric object with mirror-symmetric and mirror-asymmetric about the xyplane.
Supplementary Figure 1.(a) For an SO (2) rotationally symmetric object that is mirror-symmetric about the xy-plane, the mirror symmetry of the light-matter interaction about the xy-plane is broken by an obliquely incident (0°< θ I < 90°) linearly polarized plane wave.The lateral optical force (LOF) can be generated by the diagonally polarized light (deviates from s-or p-polarization).(b) For an SO (2) rotationally symmetric object that is mirror-symmetric about the xy-plane, the mirror symmetry of the light-matter interaction about the xy-plane is maintained when the incident angle θ I = 90°.The LOF disappears regardless of polarization orientation.(c-d) For an SO(2) rotationally symmetric object with mirror-asymmetric about the xy-plane, when the incident light is incident obliquely (0°< θ I < 90°) or vertically (θ I = 90°), LOF can be induced when the polarization angle deviates from s-or p-polarization.(e-f) The dependence of the LOF of the cylinder and hemisphere on polarization angle when the incident angle is 60° (e) and 90° (f), respectively.
In the main text, we proposed a general mirror symmetry breaking (MSB) mechanism that uses a single linearly polarized (LP) plane wave to induce MSB of light scattering and generate lateral optical force (LOF) in an SO (2) rotationally symmetric object.The MSB mentioned in our work refers to the MSB of the scattering field with respective to the xz-plane, where the particle is mirror-symmetric about the xz-plane.
Therefore, the LOF caused by the MSB is along the y-axis.The proposed mechanism only needs to satisfy the following two conditions: (1) the object has SO(2) rotational symmetry, that is, the object is mirror-symmetric about the xz-plane, (2) the oblique incidence of a diagonally polarized plane wave.To verify the MSB mechanism, six different objects with the SO(2) rotational symmetry in Fig. 1a and Fig. 2a in the main text are demonstrated to break mirror symmetry only by diagonal polarization.Here, these objects are classified into two types: (1) the mirror-symmetric object about the 3a.Here, the LOF Fy in the y direction perpendicular to the incident plane xz-plane arises from the interaction of the light scattering between the two dipoles, as illustrated in Supplementary Fig. 3b.Dipole 1 at r1 = (x0, y0, z1) and dipole 2 at r2 = (x0, y0, z2) are respectively located in semi-infinite transparent medium 1 and medium 2.An LP plane wave with an amplitude Einc polarized at angle α is incident from medium 1 to medium 2. We denote the dielectric permittivity of medium 1 and medium 2 as (ε1, ε2), respectively.
The total LOF F y acting on the two dipoles can be expressed in a general way 1   where F y di represent the LOF acting on the dipole i (i=1,2).di = αe,iEi is the induced electric dipole moment, αe,i is polarizability, Ei is the local field at the location of the dipole i.The symbol ∂y denotes the partial derivative with respect to the transversal coordinate y.The electric field Edi (r) produced by dipole di located at ri = (x0, y0, zi) can be expressed through dyadic Green's function 2 Here ω is the frequency of the incident wave, and µ0 is the vacuum dielectric permeability.
At the dipole 1 location, the local field is E1 = Einc + Eref + Ed21, as shown by the solid line in Supplementary Fig. 3b.Here, Einc and Eref are the incident field and the reflected field from the interface.Ed21 is the dipole field transmitted from dipole 2 to the location of dipole 1.Since the incident and reflected fields are independent on a lateral coordinate y, the derivative in Eq. ( 1) is ∂ y E 1 =∂ y E d21 and can be expressed with Green's functions as , .
In Eq. ( 3), G 2 tr is the transmission Green's function of the dipole 2. The derivatives can be expressed as 2 where k0 is the wavenumber in vacuum, and ε0 is permittivity in vacuum.
Similarly, for the dipole 2, the local field is E2 = Etr + Ed12, as shown by the dashed line in Supplementary Fig. 3b.Here, Etr is the optical field transmitted through the interface from the incident field Einc, and Ed12 is the dipole field transmitted from the dipole 1 to the location of the dipole 2. Since the transmitted field Etr is also independent on the lateral coordinate y, and thus the derivative in Eq. ( 1) is reduced to ∂ y E 2 = ∂ y E d12 and can be expressed as In Eq. ( 8), G Consequently, the total LOF induced by linear polarization from the interaction between the dipole scattering fields of dipole 1 and dipole 2 can be expressed as where, Supplementary Note 4: Evaluate the integral of the derivative of the Green's function.
To further understand how the incident angle θ I and polarization angle α of LP plane wave affect the magnitude and direction of the total LOF, the integral of the derivative of the Green's function should be considered.For convenience, we perform the substitutions s=k ρ /k 2 , and the Fresnel transmission coefficient t 1 p and t 2 p can be expressed as 2,3   Since the effective phase retardation between the two dipoles is mainly contributed to force, we can assume the positions of two dipoles have the relationship z 2 =- ) without loss of the generality.Using Eqs. ( 15) and ( 16) and ξ=k 2 z 1 , the integral term of Eqs.(11-14) can be rewritten as . 4 The integral term is replaced by Therefore, the final expression for the total LOF (Eq.( 10)) can be rewritten as Using the linear polarization of the incident light and Fresnel transmission coefficients for s-and p-polarization, the magnitude of the dipole moments in Eq. ( 21) are obtained  In Eq. ( 22), ts(θ I ) and tp(θ I ) are the amplitude transmission coefficients for s-and ppolarized waves, respectively.ε 12 = ε 1 /ε 2 is the relative permittivity of two media.
δ ref = 2k 1 z 1 cos θ I is the phase retardation between the incident field Einc and the reflected field Eref, and δ d12 is the phase retardation between the dipole 1 and the dipole 2. Here, δ d12 has three cases depending on the distance of the two dipoles from the interface and the incident angle, as shown in Supplementary Figs.4a-4c, where T is the refracted angle.According to the schematic diagram in Supplementary Figs.4a-4c, the three kinds of phase retardations δ d12 can be expressed as It is seen that case 1 and case 3 have the same phase retardation and it is a function of dipole position and incident angle I.The case 2 is a special case at a special angle of incidence, which satisfying tan tan Consider the case in the main text, where the incident light is incident from the air (n1 = 1) into the water (n2 = 1.337), a PS particle (np = 1.5983 @ λ = 532 nm) is semifloating at the air-water interface, the Fresnel coefficients for s-and p-polarized waves are all real values.If the dipole absorption is ignored (Im α e =0), the real part of the dipole moment terms in Eq. ( 22) can be simplified as

a) Incident from optically less dense medium ( 12 <1)
For the incident wave from the less dense medium to the denser medium, the integrals I1 and I2 have real value over the interval 0, ∞ .The integral region is divided into three intervals for discussion when the two dipoles close to (ξ→0) or far from (large ξ) the interface between two media.

a.1) For the case of the two dipoles located close to the interface (ξ→0)
In the region of integration 0, ε 12 , the transmission coefficient t 1 p s and t 2 p s are real-valued and the integral can be obtained analytically by direct integration The expressions of Re I 1 ε 12 and Re I 2 ε 12 could also be obtained by direct integration over the interval ε 12 , 1 where transmission coefficient t In the region of integration 1, ∞ , the integrand of the integrals in Eqs.(19-20) is all real-valued.The largest contribution of the integral arises from the extreme point s0 of the integrand and its immediate vicinity.In particular, s0(ε12,ξ) tends to infinity as ξ decreases to zero, and for a given ξ, the real part of integrals will converge on 1, ∞ .
It should be noted that when ξ >1, for integral I2, the integrand of the integrals has no extreme value, and the maximum value is located at the point s=1.Expand the exponential factor exp iξ ε 12 -s 2 +ε 12 1-s 2 in series at s0, and the final expression of Eq. ( 19) or Eq. ( 20) can be obtained by directly integrating each term of the series term of the integrand in I1 or I2 and f (n) s 0 is the n-order derivative of the exponential factor at s0.Here, the integrals of the first term (n=0) of the series over the interval 1, 2s 0 is exhibited, It is worth noting that one can obtain the expressions of the integral I1 and I2 by direct integration in the interval 1, ∞ for the case of high dielectric contrast (ε12<<1).
where Kn is the n-order modified Bessel function of the second kind.
Combining Eqs.(25-26) and Eqs.(33-34), the total LOF in Eq. ( 21) can be expressed as . In particular, for the case of ξ→0 4 .The Eqs. (33-34) can be further simplified as Therefore, the total LOF can be simplified as follows in the case of ξ→0 For the case of large ξ, in the interval 0, ε 12 , the asymptotic evaluation of the integral I1 and I2 can be performed by the method of stationary phase with stationary point ss = 0 5 .Performing asymptotic evaluation with accuracy up to 1/ξ 2 term, the Re I 1 ε 12 ,ξ and Re I 2 ε 12 ,ξ can be expressed as 1 In the interval ε 12 , 1 , the exponential terms of the integral can be expressed as exp -ξ s 2 -ε 12 +iξε 12 1-s 2 .One can extract the real part of the integral and perform a Taylor expansion on the trigonometric term at ε 12 since the contribution of the integral comes mainly from the lower limit of the integral.Finally, the integral can be asymptotically expanded by converting them into Laplace form 6 using the replacement of s = t 2 +ε 12 .Limiting ourselves to the terms up to 1/ξ 3 we can obtain the following expression: In the interval 1, ∞ , the integral has a sharp maximum at a point very near the lower limit of the integration, and most of the contribution to the integral arises from the immediate vicinity of this maximum.Therefore, the integral can be evaluated asymptotically by converting them into Laplace form 6 using the replacement of s = t 2 +1.Additionally, the Taylor expansion is performed at the lower limit for the decay factor exp -ξ s 2 -ε 12 .Limiting ourselves to the terms up to 1/ξ 4 Combining Eqs.(38-42) and leaving only terms to 1/ξ 2 , one can get the final asymptotic evaluation of the derivative of the Green's function: It can be seen from Eqs. (42-43) that the real part of the Green's function includes retarded       and non-oscillating 1/ξ 2 with ξ→∞.

b) Incident from optically denser medium ( 12 >1)
For the incident wave from the denser medium to the less dense medium, the integrals I1 and I2 also have real value over the interval 0, ∞ .The integral region is divided into three intervals for discussion when the two dipoles close to (ξ→0) or far from (large ξ) the interface between the two media.

b.1) For the case of the two dipoles located close to the interface (ξ→0)
In the region of 0, 1 , the integral can be obtained analytically by direct integration In the region of 1, ε 12 , the transmission coefficient is complex-valued, the real part of the derivative of Green's functions can be obtained analytically by direct In the region of ε 12 , ∞ , similar to the case of integration in the interval 1, ∞ when 12<1, the following expression can be obtained b.2) For the case of large ξ (ξ→∞) For the case of large ξ , the integral in Eqs.(19-20) can also be evaluated asymptotically by the method of stationary phase with stationary point ss = 0 in the interval 0, 1 .The integral result is the same as Eq.(38).
For the interval 1, ε 12 , the exponential terms of the integral I1 and I2 could be simplified as exp iξ ε 12 -s 2 and the integral I1 and I2 can be obtained by the method of stationary phase 5 .Using the substitution of integration variable s = t 2 +1 to avoid divergences in the endpoints, one can get     In the interval ε 12 , ∞ , the integral has a sharp maximum at a point very near the lower limit of the integration, and most of the contribution to the integral arises from the immediate vicinity of this maximum.Therefore, the integral can be evaluated asymptotically by converting them into Laplace form 6 using the replacement of s = t 2 +ε 12 .Additionally, the Taylor expansion is performed at the lower limit for the decay factor exp -ξε 12 s 2 -1 .Limiting ourselves to the terms up to 1/ξ 4 we can obtain the following expressing 22)Supplementary Figure4.(a-c) Three cases of phase retardation between the dipole 1 and the dipole 2. (d-f) The dependence of the real part of the dipole moment terms on the angle of incidence for particles with radii of 50 nm, 500 nm and 1000 nm, respectively.(g-i) The dependence of the real part of the dipole moment terms on the radius of particles at θ I = 15°, 60° and 75°, respectively.The inset shows the dual-dipole model, where the dipole radius R d =|z 1 |=|z 2 | is half of the PS particle radius R.

Supplementary Figure 5 .a. 2 )
Fig. 5a.The high fitted correlation coefficient of 0.99 confirms the R d 2 way.Supplementary Fig. 5b shows the LOF of two dipoles with a radius of 5 nm as a function of the incident angle and the force shows a gentle change that first increases and then decreases with the incident angle.a.2) For the case of large ξ (ξ→∞) Here, the dipole radius Rd is half of the PS particle radius R, which can be approximately considered equivalent to the dipole distances |z1| and |z2| away from the air-water interface (Rd =|z1|=|z2|), as shown in the dual-dipole model shown in the inset in Supplementary Fig. 4g.The oscillatory dependence of the dipole moment terms on incident angle I and dipole radius Rd is calculated and shown in Supplementary Figs.4d-4fand Supplementary Figs.4g-4i, respectively.Supplementary Figs.4d-4ffor Rd = 50 nm, 500 nm, and 1 μm show that the oscillatory variation with incident angle I become more rapid for a larger radius Rd, that is the oscillatory periodic becomes shorter for a larger radius.This is due to the cosine function of the phase Rd turns more rapid for a smaller incident angle because of the phase retardation δ d12 proportional to the term k1Rdcos(I)+k2Rdcos(T) (Rd =|z1|=|z2|) in Eq. (23).Additionally, in Supplementary Figs.4g-4i, the magnitude of the dipole terms in Eqs.(25-26) will increase with radius Rd.According to the Clausius- According to Eqs. (23-26), the phase retardations δ ref and δ d12 render the magnitude of the dipole momentum terms oscillatory with the incident angle I and dipole radius Rd. retardation and its proportion to the dipole distances |z1| and |z2| in Eqs.(22), (25) and (26).Similarly, Supplementary Figs.4g-4i for I = 15°, 60°, 75° show that the oscillatory variation with the radius I →90°), the transmission coefficients ts(θ I ) and tp(θ I ) turn zero, and the dual-dipole scattering interaction vanishes, resulting in a zero LOF.For analytical evaluation of integral I1 and I2 in Eqs.(19-21), we consider two cases of 12>1 and 12<1.